New methods in the theory of quantum spin systems

Abstract

Recently developed methods to investigate quantum spin systems are reviewed. These methods are based on somewhat unconventional applications of the spin coherent state technique developed in a series of papers in which the present authors were involved. A strict correspondence is shown to exist between energy spectra of a few spin and pseudospin systems and low-lying quantum states for a particle moving in a potential of a certain form. This suggests that quantum mechanical potential models have been found for which exact solutions are known for only a part of the spectrum (quasi-exactly solvable models). The properties of such models are discussed in detail. On the other hand, the above mentioned correspondence enables one to investigate low-temperature properties of anisotropic para- and superparamagnets. In this respect, the problem of tunnelling in such systems is of special interest. By means of the effective potential method, expressions for the tunnelling rate and metastable state decay at finite temperature are derived in a rather simple way. On a general basis an efficient potential description is developed also for some two- and many-particle systems (Dicke, Heisenberg, and Lipkin-Meshkov-Glick models). A Wigner-Kirkwood expansion of general form is constructed for spin systems of a general type. We also obtain energy quantization rules of the Bohr-Sommerfeld type with allowance for quantum corrections using an integral method and avoiding an approximate solution of the Schrödinger equation. We also discuss possible generalizations and perspectives of the methods discussed related to both group theoretical methods and applications of the quantum theory of magnetism.